what could be the problem

it sounds like possibly a grounding problem,was your alt bad when it was replaced?

Could also be a battery issue, if your battery is dying, the alternator may quit charging it when it drops below twelve volts. Most newer cars have a cut-off to avoid burning the alternator out -- the alt. recognizes input current and won't attempt to charge if the batt. is less than 12 volts.

actually in most cases the alt will continue to charge until it wears out also.

a new rebuilt alt can be bad when they check them and tell you nothing is wrong,they do check it under a load,get a new replacement and try it also have your battery checked without a load if good will be 14 volt range if not check under load,if drops belo 12volts need a battery.

Just for thoughts: 40 years ago my friend had a car that would start and run perfectly but none of the lights or anything in the car would work. We finally found that the cable from the engine to the car body was bad. Solved everything.